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Beliebt Rechnen >

integral of 1/(sin^2(x)cos^4(x))

Lösung

\int \frac{1}{sin ^{2} ( x ) cos ^{4} ( x )} d x

Lösung

- cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x) + C

Schritte zur Lösung

\int \frac{1}{sin ^{2} ( x ) cos ^{4} ( x )} d x

Trigonometrische Substitution anwenden

= \int \frac{( 1 + u ^{2} ) ^{2}}{u ^{2}} d u

Multipliziere aus

\frac{( 1 + u ^{2} ) ^{2}}{u ^{2}} : \frac{1}{u ^{2}} + 2 + u^{2}

= \int \frac{1}{u ^{2}} + 2 + u^{2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u ^{2}} d u + \int 2 d u + \int u^{2} d u

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int 2 d u = 2 u

\int u^{2} d u = \frac{u ^{3}}{3}

= - \frac{1}{u} + 2 u + \frac{u ^{3}}{3}

Setze in

u = tan (x)

ein

= - \frac{1}{tan ( x )} + 2 tan (x) + \frac{tan ^{3} ( x )}{3}

Vereinfache

- \frac{1}{tan ( x )} + 2 tan (x) + \frac{tan ^{3} ( x )}{3} : - cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x)

= - cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x)

Füge eine Konstante zur Lösung hinzu

= - cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x) + C

Graph

Beliebte Beispiele

derivative of e^xsin(x+cos(x))

\frac{d}{d x} (e^{x} sin (x) + cos (x))

limit as x approaches-5 of (5+x)^2-25

x \to - 5 lim ((5 + x)^{2} - 25)

derivative cos(4-9x^2)

d er i v a t i v e cos (4 - 9 x^{2})

derivative of (2x+5/(x^2-1))

\frac{d}{d x} (\frac{2 x + 5}{x ^{2} - 1})

derivative f(x)=2x^3+1

d er i v a t i v e f (x) = 2 x^{3} + 1